Textbook Question
Evaluate the integrals in Exercises 31–78.
49. ∫x3^(x²)dx
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.PE.133
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
Verified step by step guidance1
Understand that the problem involves radioactive decay, where the amount of carbon-14 decreases over time according to an exponential decay model.
Recall the decay formula: \(N(t) = N_0 \times e^{-\lambda t}\), where \(N(t)\) is the amount of carbon-14 remaining at time \(t\), \(N_0\) is the original amount, and \(\lambda\) is the decay constant.
Since 90% of the carbon-14 has decayed, only 10% remains, so set \(\frac{N(t)}{N_0} = 0.10\) and write the equation \(0.10 = e^{-\lambda t}\).
Take the natural logarithm of both sides to solve for \(t\): \(\ln(0.10) = -\lambda t\), which gives \(t = -\frac{\ln(0.10)}{\lambda}\).
Use the known half-life of carbon-14 (about 5730 years) to find \(\lambda\) using \(\lambda = \frac{\ln(2)}{\text{half-life}}\), then substitute \(\lambda\) back into the equation to calculate \(t\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radioactive Decay and Half-Life
Radioactive decay is the process by which unstable nuclei lose particles over time, decreasing the amount of a radioactive isotope. The half-life is the time required for half of the original radioactive atoms to decay. Understanding half-life allows us to relate the remaining amount of a substance to the elapsed time.
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Exponential Growth & Decay
Carbon-14 Dating Method
Carbon-14 dating estimates the age of organic materials by measuring the remaining carbon-14 isotope, which decays at a known rate. Since living organisms constantly replenish carbon-14, the decrease after death indicates elapsed time. This method is widely used in archaeology and geology.
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Exponential Decay Formula
The exponential decay formula, N(t) = N_0 * (1/2)^(t/T), relates the remaining quantity N(t) to the initial amount N_0, elapsed time t, and half-life T. This formula helps calculate the age of a sample when the fraction of remaining radioactive material is known.
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Related Practice
Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
f. f(x) = sech(x), g(x) = e^(-x)
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
c. f(x) = 10x^3 + 2x^2, g(x) = e^x
Textbook Question
In Exercises 79–84, solve for y.
79. 3^y = 2^(y+1)
Textbook Question
113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).